Note that I'm not really covering static screen capture here, since short cuts are built into each operating system for static capture.

iOS is pretty locked down. And without jail breaking, your options are limited.

However, iOS has a built in screen sharing capability called AirPlay, designed to be used for streaming to your Apple TV. But that hasn't stopped some enterprising developers building AirPlay servers for both Windows and Mac OS computers.

That would be largely missing the point of doing this. With these tools you capture the real output from a real world, not a simulated device. Using a simulator and finding defects may find issues not present on a real device, and likewise what you capture via the simulator may not match the experience on a real device as well.